A Note on Antenna Selection in Gaussian MIMO Channels: Capacity Guarantees and Bounds

We consider the problem of selecting $k_t \times k_r$ antennas from a Gaussian MIMO channel with $n_t \times n_r$ antennas, where $k_t \leq n_t$ and $k_r \leq n_r$. We prove the following two results that hold universally, in the sense that they do not depend on the channel coefficients: (i) The capacity of the best $k_t \times k_r$ subchannel is always lower bounded by a fraction $\frac{k_t k_r}{n_t n_r}$ of the full capacity (with $n_t \times n_r$ antennas). This bound is tight as the channel coefficients diminish in magnitude. (ii) There always exists a selection of $k_t \times k_r$ antennas (including the best) that achieves a fraction greater than $\frac{\min(k_t ,k_r)}{\min(n_t,n_r)}$ of the full capacity within an additive constant that is independent of the coefficients in the channel matrix. The key mathematical idea that allows us to derive these universal bounds is to directly relate the determinants of principle sub-matrices of a Hermitian matrix to the determinant of the entire matrix.